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The finite-region thermal modular Hamiltonian as the missing tool: an open problem for the independent relative-entropy source near a Schwarzschild horizon

Miguel Ángel Martínez Martí

Published Jun 26, 2026• 0 citations
Publisher

Abstract

This preprint poses, and does not solve, an open problem concerning the relative-entropy source near a Schwarzschild horizon. The central object is the finite-region thermal modular Hamiltonian on the Schwarzschild exterior: the modular Hamiltonian of a finite ball at finite radius in the Hartle–Hawking (KMS) state. Its strictly near-horizon, conformal limit is known — it is the Cardy–Tonni thermal-interval Hamiltonian of the two-dimensional s-wave reduction — but its finite-distance form, away from that limit, is not. Two independent obstructions force this object. First, the second-order Blanco–Casini–Hung–Myers stress-tensor factorisation of the relative entropy, valid near the vacuum, degrades once the ball is large compared with the local (Tolman-redshifted) thermal length. Second, the squeezed-state coincidence wall survives the Type II crossed-product renormalization as the Jafferis–Lewkowycz–Maldacena–Suh invariant. The two obstructions are logically independent. In the two-dimensional s-wave reduced sector, the finite content that the missing operator would govern is exhibited in closed form for non-squeezed states (where the wall is absent): a finite, cutoff-independent remainder R[σ] lying below the wall. In four dimensions, the operator itself is absent from the canonical relevant-deformation and modular-Hamiltonian toolkit — existing constructions either compute entanglement entropy from an already-known modular Hamiltonian, or supply an operator-level first-order deformation only for a vacuum reference, and neither applies to a finite-region thermal KMS reference. The paper therefore states, as its open problem, the construction of this operator — or its first-order deformation off the conformal limit — and claims no resolution of the squeezed-state wall. This work is part of the Informational Gravity Framework series and continues the capstone deposit (concept DOI 10.5281/zenodo.20804927).

DOI
10.5281/zenodo.20870916

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This preprint poses, and does not solve, an open problem concerning the relative-entropy source near a Schwarzschild horizon. The central object is the finite-region thermal modular Hamiltonian on the Schwarzschild exterior: the modular Hamiltonian of a finite ball at finite radius in the Hartle–Hawking (KMS) state. Its strictly near-horizon, conformal limit is known — it is the Cardy–Tonni thermal-interval Hamiltonian of the two-dimensional s-wave reduction — but its finite-distance form, away from that limit, is not. Two independent obstructions force this object. First, the second-order Blanco–Casini–Hung–Myers stress-tensor factorisation of the relative entropy, valid near the vacuum, degrades once the ball is large compared with the local (Tolman-redshifted) thermal length. Second, the squeezed-state coincidence wall survives the Type II crossed-product renormalization as the Jafferis–Lewkowycz–Maldacena–Suh invariant. The two obstructions are logically independent. In the two-dimensional s-wave reduced sector, the finite content that the missing operator would govern is exhibited in closed form for non-squeezed states (where the wall is absent): a finite, cutoff-independent remainder R[σ] lying below the wall. In four dimensions, the operator itself is absent from the canonical relevant-deformation and modular-Hamiltonian toolkit — existing constructions either compute entanglement entropy from an already-known modular Hamiltonian, or supply an operator-level first-order deformation only for a vacuum reference, and neither applies to a finite-region thermal KMS reference. The paper therefore states, as its open problem, the construction of this operator — or its first-order deformation off the conformal limit — and claims no resolution of the squeezed-state wall. This work is part of the Informational Gravity Framework series and continues the capstone deposit (concept DOI 10.5281/zenodo.20804927).
This work analyzes the degradation of continuous-variable entanglement between an inertial reference mode (Alice) and a mode accessible to a noninertial observer (Rob) when horizon-induced mode splitting renders part of Rob's degrees of freedom inaccessible. Within an effective single-mode Gaussian description, tracing the inaccessible sector induces on Rob a local phase-insensitive Gaussian amplifier with gain g = cosh²r (quantum-limited when the environment is in vacuum), where r is the Unruh squeezing parameter. For a standard-form two-mode input we provide closed-form output covariance matrices and the logarithmic negativity E_N, and we derive an exact threshold separating (i) a persistent large-r exponential tail E_N ~ C·exp(−2r) from (ii) finite-r sudden death when additional thermal noise is present. In the quantum-limited pure-input case the prefactor is C(s) = 8·tanh²(s)/ln 2. We further show that, within this Gaussian truncation, the asymptotic scale of the entanglement tail is controlled by the effective modular temperature β_R of Rob's reduced single-mode state. All quantitative claims are verified numerically. Connections to type-III algebraic modular theory and to semiclassical gravitational entropy are discussed as a delineated program rather than derived results; in particular, no identification of C(s) with A/4G or with the Bekenstein–Hawking entropy is claimed. A falsifiable analogue-gravity prediction (BEC or optical horizons) is given in terms of the time decay of the Hawking–partner logarithmic negativity. This preprint develops quantitatively a mechanism from the author's earlier work, "Incomplete Causal Transactions: Black Holes as Permanently Frustrated Couplers" (DOI: 10.5281/zenodo.18736657). AI use disclosure: AI systems were used as tools for symbolic and numerical verification of the Gaussian calculations and for reference checking; all quantitative claims were independently verified by direct computation, and the author takes full responsibility for the work and its delimited scope.
A bstract Recently, it has been observed that the Hartle-Hawking correlators, a signature of smooth horizon, can emerge from certain heavy excited state correlators in the (manifestly non-smooth) BTZ stretched horizon background, in the limit when the stretched horizon approaches the real horizon. In this note, we develop a framework of quantizing the CFT modular Hamiltonian, that explains the necessity of introducing a stretched horizon and the emergence of thermal features in the AdS-Rindler and (planar) BTZ backgrounds. In more detail, we quantize vacuum modular Hamiltonian on a spatial segment of S 1 , which can be written as a particular linear combination of sl(2,ℝ) generators. Unlike radial quantization, (Euclidean) time circles emerge naturally here which can be contracted smoothly to the ‘fixed points’(end points of the interval) of this quantization thus providing a direct link to thermal physics. To define a Hilbert space with discrete normalizable states and to construct a Virasoro algebra with finite central extension, a natural regulator ( ϵ ) is needed around the fixed points. Eventually, in the dual description the fixed points correspond to the horizons of AdS-Rindler patch or (planar) BTZ and the cut-off being the stretched horizon. We construct a (Lorentzian) highest weight representation of that Virasoro algebra where vacuum can be identified with certain boundary states on the cut-off surface. We further demonstrate that two point function in a (vacuum) descendant state of the regulated Hilbert space will reproduce thermal answer in ϵ → 0 limit which is analogous to the recent observation of emergent thermality in (planar) BTZ stretched horizon background. We also argue the thermal entropy of this quantization coincides with entanglement entropy of the subregion. Conversely, the microcanonical entropy corresponding to high energy density of states exactly reproduce the BTZ entropy. Quite remarkably, all these dominant high lying microstates are defined only at finite ϵ in the regulated Hilbert space. We expect that all our observations can be generalized to BTZ in stretched horizon background where the boundary spatial coordinate is compactified.
Abstract The modular Hamiltonian of reduced states, given essentially by the logarithm of the reduced density matrix, plays an important role within the AdS/CFT correspondence in view of its relation to quantum information. In particular, it is an essential ingredient for quantum information measures of distances between states, such as the relative entropy and the Fisher information metric. However, the modular Hamiltonian is known explicitly only for a few examples. For a family of states ρ λ that is parametrized by a scalar λ, the first order contribution in λ∼=λ−λ0 \overset{\sim }{\lambda } = \lambda -{\lambda}_0 λ∼=λ−λ0​ λ ∼ = λ − λ 0 of the modular Hamiltonian to the relative entropy between ρλ and a reference state ρλ 0 is completely determined by the entanglement entropy, via the first law of entanglement. For several examples, e.g. for ball-shaped regions in the ground state of CFTs, higher order contributions are known to vanish. In these cases the modular Hamiltonian contributes to the Fisher information metric in a trivial way. We investigate under which conditions the modular Hamiltonian provides a non-trivial contribution to the Fisher information metric, i.e. when the contribution of the modular Hamiltonian to the relative entropy is of higher order in λ∼ \overset{\sim }{\lambda } λ∼ λ ∼ . We consider one-parameter families of reduced states on two entangling regions that form an entanglement plateau, i.e. the entanglement entropies of the two regions saturate the Araki-Lieb inequality. We show that in general, at least one of the relative entropies of the two entangling regions is expected to involve λ∼ \overset{\sim }{\lambda } λ∼ λ ∼ contributions of higher order from the modular Hamiltonian. Furthermore, we consider the implications of this observation for prominent AdS/CFT examples that form entanglement plateaux in the large N limit.
We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1)U(1)U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.
Phase IV closed the free/geometric-flow sector on the third verdict: the registered modular orientation is geometric and irreducible to any CPTP channel, yet it coincides with the Borchers–Wiesbrock spectral condition (energy positivity) — already-known QFT serving as the arrow. That verdict rests on a single structural fact: a KMS state has one modular temperature (Tomita–Takesaki). Any genuinely new arrow beyond the spectral condition therefore requires a state that is not KMS nor generalized-Gibbs. This paper attacks the one corner that is at once outside the third verdict, in scope (no gravity), and tractable: the current-carrying non-equilibrium steady state (NESS). Proceeding symmetrically, with preregistered decision rules frozen before any run: Island A (a Generalized Gibbs Ensemble after an integrable quench) closes null / horn b — the charge tower is absorbed by a single recalibration c(R) → 1, and a finite-size positive is diagnosed as a charge-basis conditioning artefact. The genuine NESS is decided in two stages. On the lattice, the preregistered instrument returns HORN A, grade (ii): the bulk-interval modular Hamiltonian is irreducible to the local-equilibrium-plus-current generator, with a clean law r∞ ≈ Δβ/2. A second, independently frozen preregistration takes the field limit a/β → 0 and returns global collapse: the decisive residual against the Bernard–Doyon boost-of-thermal reference vanishes as ~a² to the equilibrium floor across every drive, while the discriminator's teeth stay intact. The lattice HORN A is reclassified as a UV band-curvature (inflection-point) artefact; the growing distance-to-local-cone that produced it measures the intrinsic non-locality of the Cardy–Tonni boost-of-thermal entanglement Hamiltonian, a known-QFT feature, not new structure. The third verdict extends to the continuum non-KMS, positive-entropy-production sector. Any genuine new modular arrow is pushed to strongly-coupled / holographic NESS, named and left out of scope.
The mean Hawking flux does not determine how quantum stress fluctuations perturb a black-hole geometry. That information resides in field correlations, including vacuum contributions absent from the outgoing luminosity. Here we connect the complete exterior Unruh Wightman function to a finite-band stochastic reconstruction of Schwarzschild geometry. We retain the incoming vacuum and both outgoing frequency branches, form the connected stress covariance before angular projection, and propagate its monopole through a density-driven radial response. A shared Gaussian reconstruction then resolves how the value and derivative of the lapse control different geometric observables. For the specified covariance and noise amplitude, 2,200 realizations give relative widths of 0.0185 for the local horizon radius and 0.0483 for the surface-gravity temperature parameter, consistently across histograms and reconstruction checks. Covariance derivatives explain this unequal sensitivity. The construction separates the quantum-state input from the gravitational closure and spatial sampling assumptions, providing a reproducible finite-resolution description of how Hawking-field correlations affect horizon positions and surface-gravity statistics.